Chataur in 26 described string topology in terms of the geometric cycles approach to homology 58. These notes grew out of a graduate topology course i gave at stanford university during the spring term, 1998. Stuart hall, emeritus professor, the open university. If xo y are homotopy equivalences, then the induced maps t.
The purpose of this paper is to develop walls finiteness obstruction wal, wa2 from an extremely geometrical point of view. Grothendiecks problem homotopy type theory synthetic 1groupoids category theory rethinking the homotopy hypothesis old the analytic homotopy theory of 1groupoids is equivalent to that of spaces. Introduction to higher homotopy groups and obstruction theory michael hutchings february 17, 2011 abstract these are some notes to accompany the beginning of a secondsemester algebraic topology course. Siebenmanns infinite simple homotopy theory s and hatchets higher simple homotopy theory h are developed along these geometric lines. Equivariant stable homotopy theory 5 isotropy groups and universal spaces. Second, the analog of theorem 2 for shape theory is false. Let us mention three additional approaches to string topology. A course in simplehomotopy theory marshall m cohen springer. In, which is a textbook treatment of this and many other aspects of classical homotopy theory, fred gives no less than 5 equivalent formulations of this question. This is an introduction to type theory, synthetic topology, and homotopy type theory from a categorytheoretic and topological point of view, written as a chapter for the book new spaces for mathematics and physics ed.
In the 30s50s, much of homotopy theory was focused on developing combinatorial ways to determine homotopy equivalences. Cohen department of mathematics stanford university. Introduction to higher homotopy groups and obstruction theory. In generality, homotopy theory is the study of mathematical contexts in which functions or rather homomorphisms are equipped with a concept of homotopy between them, hence with a concept of equivalent deformations of morphisms, and then iteratively with homotopies of homotopies between those, and so forth. In topology, two continuous functions from one topological space to another are called homotopic from greek. We can rst reduce to the case of a single elementary expansioncollapse, in which case the relative. Pdf on jan 1, 1995, ran levi and others published on finite groups and homotopy. We present a new approach to simple homotopy theory of polyhedra. A simple observation due to bousfield and kan 1 shows that for any finite. Illustrated there is a regular cw complex x along with a simple. This follows by the previous paragraph and the vanishing of. Simple homotopy types and finite spaces jonathan ariel barmak and elias gabriel minian abstract. In 2 we describe how we shall attempt to formulate homotopy theory in a particularly simple way. The topic can also serve as an introduction to algebraic k theory from a topological viewpoint.
The applications are to differentiable manifolds, pl manifolds and topological manifolds. Sections 7 and 8 describe in detail two basic examples of model categories, namely the category top of topological spaces and the category ch rof nonnegative chain. The proposal was to attach a stable homotopy type to the geometric data given in. Two cwcomplexes are simple homotopy equivalent if they are related by a sequence of collapses and expansions inverses of collapses, and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map.
An elementary move in this setting is called a strong. Cohen 2 suggests that if g is a finite superperfect group then. Cohen in 27 studies string topology from the viewpoint of morse theory, and shows that the floer homology. Your print orders will be fulfilled, even in these challenging times. A simplehomotopy approach to the finiteness obstruction. A course in simplehomotopy theory graduate texts in. The arccomponents of bc are, by cohen s theorem, exactly the simple homotopy types of finite polyhedra. A course in number theory and cryptography, neal koblitz. A course in simple homotopy theory, springerverlag, berlin and new york. The two basic examples which arise in floer theory are the following. This book contains a semihistorical and geometrically motivated exposition of j. Springer have made a bunch of books available for free, here.
Simple homotopy theory and whitehead torsion richard wong abstract. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple homotopy theory. In chapter 2, the theory of framed cobordism was discussed to calculate stable homotopy groups of spheres. Strong homotopy types, nerves and collapses jonathan ariel barmak and elias gabriel minian abstract. This is an excellent subject for a topology student to grow up on. In the end many pages hence this attempt fails, but the theory which has been created in the meantime turns out to be rich and powerful in its own right. A to describe some of our ideas concerning what, exactly, the floer homology groups compute. In 1995 the author, jones, and segal introduced the notion of floer homotopy theory. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. For the proof of the 2 cohen macaulayness we will derive some remarkable results about the homotopy type of posets, in particular on posets. Many great works has been done for the last 90 years, but there are much more we want to know and want to understand. Floers infinite dimensional morse theory title and. This is a lecture on a historical motivation for looking at algebraic k theory. On the cohenmacaulay connectivity the homotopy type of.
On the cohenmacaulay connectivity the homotopy type of posets. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simplehomotopy theory. We present a new approach to simple homotopy theory of polyhedra using. Siebenmanns infinite simple homotopy theory s and hatchets higher simple homotopy theory h are. A course in simplehomotopy theory, graduate texts in mathematics, no. Cohen, a course in simplehomotopy theory, graduate texts in math. One such danger is a ready susceptibility to simple explanations. Bck is a union of components of bc, those containing polyhedra homotopy equivalent to k. A second reason for writing the book is pedagogical. Sorry, we are unable to provide the full text but you may find it at the following locations. In mathematics, particularly the area of topology, a simple homotopy equivalence is a refinement of the concept of homotopy equivalence.
Furthermore, the homomorphism induced in reduced homology by the inclusion xr. They form the rst four chapters of a book on simplicial homotopy theory, which we are currently preparing. These products give the homotopy theory of loop spaces the structure of graded lie algebras with the exception of some unfortunate failure of the jacobi identity at the prime 3. The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories e.
Pdf lecture notes in algebraic topology semantic scholar. A course in simplehomotopy theory marshall m cohen. Recall from whiteheads simple homotopy theory the basic geometric. Finally, we state the full form of the hurewicz theorem without proof. In more detail, the contents of the two parts of the thesis are as follows. The goal is to introduce homotopy groups and their uses, and at the same time to prepare a bit for the. Stable homotopy theory is homotopy theory in the case that the operations of looping and delooping are equivalences as homotopy theory is the study of homotopy types, so stable homotopy theory is the study of stable homotopy types. A geometric approach to homotopy theory springerlink. A homotopy type is an equivalence class of homotopy equivalent spaces. Cohen department of mathematics stanford university bldg. A course in simplehomotopy theory graduate texts in mathematics this book grew out of courses which i taught at cornell university and the university of warwick during 1969 and 1970. Kamps, tim porter, abstract homotopy and simple homotopy theory, world scientific 1997. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by cohen, moore, and the author, on the exponents of homotopy groups. Chain complexes, homology, and cohomology homological algebra products fiber bundles homology with local coefficients fibrations, cofibrations and homotopy groups obstruction theory and eilenbergmaclane spaces bordism, spectra, and generalized homology spectral sequences further applications of spectral sequences simple homotopy theory bibliography.
Homotop y equi valence is a weak er relation than topological equi valence, i. Even if our prediction was confirmed, that is, if autistic children lacked a theory of mind, we would still have to establish that this was a specific deficit, that is, largely. Some of his famous problems were on number theory, and have also been in. As voevodskys work became integrated with the community of other researchers working on homotopy type theory, univalent foundations was sometimes used interchangeably with homotopy type theory, and other times to refer only to its use as a foundational system excluding, for example, the study of modelcategorical semantics or.
Floers infinite dimensional morse theory and homotopy theory. Discrete morse theory and localization vidit nanda abstract. He proved the fundamental theorems of abelian class. The seminar should be of interest to those interested in algebraic topology. The topic can also serve as an introduction to algebraic ktheory from a topological viewpoint. Preceding the four main chapters there is a preliminary chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in both the homological and homotopical sides of. Pdf on finite groups and homotopy theory researchgate. Algebraic methods in unstable homotopy theory this is a comprehensive uptodate treatment of unstable homotopy. Whiteheads theory of simple homotopy types is inspired by tietzes theorem in combinatorial group theory, which states that any. A more abstract, but at the same time geometric, approach to simple homotopy theory was explored in cohens book as well as in the papers by eckmann, eckmann and maumary, and siebenmann, listed above. In fact, the zeeman conjecture, which states that k. The whitehead torsion is important in applying surgery theory to nonsimply connected manifolds of dimension 4.
Here is the homotopy type of the most simple space. Printed in great britain homotopy, simple homotopy and compacta steve ferryi received 5 may 1978. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. When good references are available we may not include the details of all the proofs. I have tried to write proofs which meet the needs of such students. Depending on the interest in the topic, there may be a continuation of the seminar next semester about geometric applications of simple homotopy theory and connections with differential topology e. A notable use of homotopy is the definition of homotopy groups and. I am very grateful to the students in that course for comments on earlier versions of these notes. We introduce the theory of strong homotopy types of simplicial complexes. I is collapsible for any contractible polyhedron kof dimension 2, is still an open problem. A course in simple homotopy theory, graduate texts in mathematics, no. Homotopy equivalence is an equivalence relation on spaces. The proposal was to attach a stable homotopy type to the geometric data given in a version of floer. New the synthetic theory of 1groupoids ismodeled byspaces.
If you dont want to wait have a look at our ebook offers and start reading immediately. For our purposes the \ homotopy theory associated to c is the homotopy category hoc together with various related constructions x10. Also, r0 wh k is just the group called wh k in 81, where white. Introduction to combinatorial homotopy theory francis sergeraert ictp map summer school august 2008 1 introduction. My research interests are mainly in topology with connections to other areas such as. A course in some aspects of classical homotopy theory. A space is contractible if it is homotopy equivalent to a onepoint space. But above all i wish to thank fred himself for simply being there, and doing. In particular, they were interested in when a homotopy equivalence can be. Finally, we will discuss lens spaces which provide examples of homotopy equivalent nite cw complexes which are not simple homotopy equivalent. Homotopy, simple homotopy and compacta sciencedirect. These notes were used by the second author in a course on simplicial homotopy theory given at the crm in february 2008 in preparation for the advanced courses on simplicial methods in higher categories that followed.
Element ar y homo t opy theor y homotop y theory, which is the main part of algebraic topology, studies topological objects up to homotop y equi valence. In mathematical logic and computer science, homotopy type theory hott h. The idea of homotopy can be turned into a formal category of category theory. Introduction homotopy equivalence whitehead s combinatorial approach to homotopy theory cw complexes a geometric. Part ii of this thesis concerns a computational problem in parametrized stable homotopy, namely the determination of the twisted khomology of the simple lie groups. Finally, i provided a brief summary of the extremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme interactions between homotopy theory and algebra. Incidence relations among the cells of a regular cw complex produce a poset enriched category of entrance paths whose classifying space is homotopy equivalent to that complex.
Simple homotopy types of homcomplexes, neighborhood. Cohen s approach has proven to be very influential. A survey of computations of homotopy groups of spheres. The goal is to understand homotopy theory geometrically.